Lecture 20: superimposed strain representation of strain

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Lecture 20 superimposed strain representation
of strain strain measurement
websites from which images are drawn
http//www.leeds.ac.uk/learnstructure/index.htm h
ttp//www.rci.rutgers.edu/geolweb/slides.html
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what happened to these trilobites?
they were distortedexperienced permanent shape
changes (not elastic!) study of these
distortions is deformation analysis remember
STRAIN
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think back in homogeneous strain, circles
become ellipses (2D)
what does this mean? circles correspond to
undeformed state ellipses correspond to
deformed state
comparison of circle to ellipse yields strain
ellipse is the strain ellipse
note ellipses above
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what does the strain ellipse tell us?
circle extends beyond ellipse area of
shortening
ellipse extends beyond circle area of
extension
ellipse extends beyond circle area of
extension
circle extends beyond ellipse area of
shortening
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strain change in length of lines from circle to
ellipse
colors of undeformed and deformed lines denote
same line in the two different states
(before deformation and after) magenta line
shortened during deformation pink line
lengthened during deformation
from http//www.leeds.ac.uk/learnstructure/index.
htm
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back to strain ellipse and unit circle
where e1 gt 0 and e2 lt 0
one set of lines does not change in
length marked by intersection of circle and
ellipse
what are lines?material lines connect features,
such as grains, that are recognizable throughout
the bodys strain history no contrast between
material line and body---passive markers
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back to undeformed and deformed lines
in diagram below, note that blue and
purple lines are perpendicular to each other

both in ellipse and in circle
other pairs of lines do not maintain
angular relationship with each other e.g.
green-pink angle gets smaller
for 2D homogeneous strain, only two orientations
of material lines remain perpendicular
before and after strain
form major and minor axes of strain ellipse
lines change in length (otherwise no ellipse
forms)
in 3D ellipsoid with axes X, Y, and Z as the
principal strain axes X Y Z X
maximum Z minimum Y intermediate
(for 2D ellipse, X and Z are used)
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finite strain measure of strain that
compares initial and final configuration
independent of intermediate steps between
initial and final stages
intermediate strain steps may be determined
separately intermediate steps called
incremental strains
the net effect of all incremental strains
produces finite strain
many ways to measure finite strain in rock
incremental strain is more difficult to
determine, but may be more important for
deformation history
series of incremental strains generates a strain
path finite strain may be the same, but
strain path is different
when incremental strains become very small
infinitesimal strains
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strain path series of incremental
(infinitesimal) strains to achieve finite strain
several possibilities to get from initial to
final state
initial
final
in upper path, shortening always vertical in
lower path, shortening horizontal first vertical
second
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all material lines in the body except the
principal strain axes (which remain
perpendicular to each other through
deformation) rotate relative to each other
in some cases, the material lines that are
perpendicular before and after each strain
increment are not the same throughout strain
history. e.g. these material lines rotate
relative to finite strain axes
this is noncoaxial strain accumulation
in other cases, same material lines remain
perpendicular throughout and do not rotate
relative to finite strain axes
this is coaxial strain accumulation
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noncoaxial strain principal strain axes rotate
simple shear noncoaxial, constant-volume, 2D
deformation (slip within deck of cards)
coaxial strain principal strain axes do not
rotate
pure shear coaxial, constant-volume deformation
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we will now do some simple experiments and
examine strain path and nonrotation/rotation of
strain axes
several possibilities to get from initial to
final state
initial
final
in upper path, shortening always vertical in
lower path, shortening horizontal first vertical
second
what does this mean for incremental strains?
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each incremental strain has its own incremental
strain ellipse
finite strain ellipse from previous slide
total strain is vertical shortening
incremental strain ellipse from upper path, first
step
vertical shortening as in finite strain ellipse
lines shorten (or extend) throughout
incremental strain ellipse from lower path, first
step horizontal shortening differs from
finite strain ellipse lines
originally extending will later shorten
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in figure below, a coaxial incremental
strain ellipse of different magnitude
is superimposed on the first ellipse
extension
incremental ellipse
undeformed circle


shortening
from van der Pluijm and Marshak, 1997
three regions occur in the subsequent ellipse
(sum of two) I region of continued
extension II region of continued
shortening III region of initial shortening
that is now extension
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in figure below, a noncoaxial incremental
strain ellipse of different magnitude
is superimposed on the first ellipse
extension
incremental ellipse
undeformed circle


shortening
geometry is more complex four regions occur
in the subsequent ellipse (sum of two) I
region of continued extension II region of
continued shortening III region of initial
shortening that is now extension IV
region of initial extension that is now shortening
from van der Pluijm and Marshak, 1997